PARADISE LOST? efficient satisfies a1"2 - 2x -j- 10 = 0, the second, x- - 4x — 421 = 0, the third, a? — 90 = 0, of the respective degrees 2.2,3;. Lnagine (if you can) the set of all algebraic numbers. Among these will be all the positive rational integers 1,2, 3, ... , since any one of them, say n, satisfies an algebraic equation, x — 72 = 0, in which the coefficients (1, and — n) are rational integers. But in addition 10 these the set of all algebraic numbers will include all roots of all quadratic equations with rational integer coefficients, and all roots of all cubic equations with rational integer coefficients, and so on, indefinitely. Is it not intuitively evident that the set of all algebraic numbers will contain infi- nitely more members than its sw&-set of the rational integers 1. 2, 3, ... ? It might indeed be so, but it happens to be false. Cantor proved that the set of all rational integers 1, 2, 3, ... contains precisely as many members as the Infinitely more inclusive' set of all algebraic numbers. A proof of this paradoxical statement cannot be given here, but the kind of device - that of 'one-to-one correspondence* - upon which the proof is based can easily be made intelligible. This should induce in the philosophical mind an understanding of what a cardinal number is. Before describing this simple but somewhat elusive concept it will be helpful to glance at an expression of opinion on this and other definitions of Cantor's theory which emphasizes a distinction between the attitudes of some mathematicians and many philosophers toward all questions regarding 'number* or 'magnitude". ;A mathematician never defines magnitudes in themselves, as a philosopher would be tempted to do; he defines their equality, their sum, and their product, and these definitions determine, or rather constitute, all the mathematical properties of magnitudes. In a yet more abstract and more formal manner he lays dawn symbols and at the same time prescribes the rules according to which they must be combined; these rules suffice to characterize these symbols and to give them a mathematical value. Briefly, he creates mathematical entities by means of arbitrary conventions, in the same way that the several chess* men are defined by the conventions which govern their moves 623